Refined Plate Theory Research Articles

Free vibration, buckling and bending analyses of multilayer functionally graded graphene nanoplatelets reinforced composite plates using the NURBS formulation

In this study, a NURBS formulation based on the four-variable refined plate theory (RPT) for free vibration, buckling and static bending analyses of multilayer functionally graded graphene platelets reinforced composite (FG GPLRC) plates, for the first time, is proposed. The distributions of graphene platelets (GPLs) in the polymer matrix either uniformly or non-uniformly including different patterns are considered. The Young’s modulus of the nanocomposites is predicted by the modified Halpin–Tsai model, while the Poisson’s ratio and density mass are implemented by the rule of mixtures. Governing equations are derived and the NURBS formulation is employed to obtain natural frequencies, critical buckling loads and deflections of multilayer FG GPLRC plates. Thanks to continuous higher-order derivatives of NURBS basis functions in isogeometric analysis (IGA), the present approximation is easy to satisfy the C1-continuty requirement of the RPT model. In addition, a rotation-free technique is applied to eliminate the bending and shear slopes in the case of clamped boundaries. Effects played by GPLs weight fraction, GPLs distribution patterns, number of layers, thickness-to-length ratio are investigated. Numerical results indicate that the inclusion of GPLs can significantly improve the stiffness of plates.

Porosity and inhomogeneity effects on the buckling and vibration of double-FGM nanoplates via a quasi-3D refined theory

This study is devoted to illustrate the mechanical buckling and free vibration analyses of double-porous functionally graded (FG) nanoplates embedded in an elastic foundation. A new quasi-3D refined plate theory is presented to model the displacement field. This theory contains only five unknown functions and considers the shear strain as well as thickness stretching. Based on the modified Mooney-type exponential relation, a new exponential law is presented to govern the materials variation and porosities distribution through the thickness of the nanoplates. The two porous nanoplates are bonded together by a set of parallel elastic springs and surrounded by Pasternak medium. The nonlocal strain gradient theory containing the nonlocal parameter and gradient coefficient is utilized to study the size-dependent effects. Based on Hamilton’s principle, the equations of motion are drawn including the material parameters, elastic foundation reaction and biaxial compressive forces. An analytical approach for simply-supported and clamped bilayer porous FG nanoplates is implemented. The obtained results are compared with those available in the literature. Additional numerical calculations are introduced to show the influences of the material length scale parameters, inhomogeneity parameter, porosity factor and other parameters on the critical buckling and frequencies of the double-porous FG nanoplates.

A nonlocal strain gradient refined plate theory for dynamic instability of embedded graphene sheet including thermal effects

In this study, nonlocal strain gradient theory (NSGT) is applied to examine the dynamic instability of embedded viscoelastic graphene sheet under periodic axial load including thermal effects. The foundation is simulated by visco-Pasternak model containing springs, dampers and a shear layer. The motion equations are derived according to the four-variable refined shear deformation plate theory and via Hamilton’s principle. The equations are converted into a linear system of Mathieu-Hill equations by means of Navier’s method. Afterwards, Bolotin’s approach is utilized to determine the principle unstable region of graphene sheet. The influences of nonlocal parameter, structural damping coefficient, length scale parameter, static load factor, temperature variation, foundation type as well as aspect ratio on the dynamic stability of graphene sheet are investigated. Based on the numerical results, it is indicated that with enlarging the nonlocal parameter, static load factor and temperature change, the excitation frequency decreases and so, instability region shifts to left side while the effect of length scale parameter is on the contrary. Additionally, it is indicated that when the length scale parameter enhances, the effects of temperature and foundation on the instability region of graphene sheet reduce.

Vibration of cracked functionally graded microplates by the strain gradient theory and extended isogeometric analysis

In this study, the vibration behaviours of functionally graded microplates with cracks are investigated by means of a simple yet rigorous version of Mindlin’s generialised continuum and the extended isogeometric analysis (XIGA). The simplified strain gradient theory which includes one material length parameter and an additional micro-inertia term is employed to capture the size effects. Meanwhile, the displacement field of the plates is described using the refined plate theory with four unknowns and the XIGA in which enrichment functions are involved to effectively predict the responses of microplates with cracks. In addition, the IGA approach with highly smooth basis functions of non-uniform rational B-spline (NURBS) ensures a clean and efficient treatment of higher continuity requirements in the strain gradient theory. The benchmark numerical results show significant departure from those analysed by the classical continuum elasticity. Indeed, they reveal strong influences of microstructural characteristics on the vibration responses of microplates which are not shown in the platform of the classical theory and the influences are more pronounced as the size of the plates becomes comparable with the material length parameter.

Bending and buckling of three-dimensional graphene foam plates

Abstract Bending and buckling behaviors of three-dimensional graphene foam (3D-GrF) plates are investigated in the framework of the two-variable refined plate theory (TRPT). The material properties of three-dimensional graphene foams vary continuously along thickness direction of the plate. The TRPT considers the effect of transverse shear strains that vary quadratically along the thickness of the plate. In addition, this theory does not need to consider the shear correction factor. By using Hamilton’s principle, the governing equations are obtained. Analytical solutions for the bending problem of 3D-GrF plates are derived via Navier’s method. Additionally, buckling problem of 3D-GrF plates with various boundary conditions are analyzed by employing the Galerkin method. Results obtained in this study are verified by comparing with the published ones. Finally, the influences of pore distribution, porosity coefficient, transverse distributed load, aspect ratio, length-to-thickness ratio, mode number and axial compression ratio on bending and buckling behaviors of 3D-GrF plates are discussed.

Assessment of porosity influence on dynamic characteristics of smart heterogeneous magneto-electro-elastic plates

A four-variable shear deformation refined plate theory has been proposed for dynamic characteristics of smart plates made of porous magneto-electro-elastic functionally graded (MEE-FG) materials with various boundary conditions by using an analytical method. Magneto-electro-elastic properties of FGM plate are supposed to vary through the thickness direction and are estimated through the modified power-law rule in which the porosities with even and uneven type are approximated. Pores possibly occur inside functionally graded materials (FGMs) due the result of technical problems that lead to creation of micro-voids in these materials. The variation of pores along the thickness direction influences the mechanical properties. The governing differential equations and boundary conditions of embedded porous FGM plate under magneto-electrical field are derived through Hamilton\'s principle based on a four-variable tangential-exponential refined theory which avoids the use of shear correction factors. An analytical solution procedure is used to achieve the natural frequencies of embedded porous FG plate supposed to magneto-electrical field with various boundary condition. A parametric study is led to carry out the effects of material graduation exponent, coefficient of porosity, magnetic potential, electric voltage, elastic foundation parameters, various boundary conditions and plate side-to-thickness ratio on natural frequencies of the porous MEE-FG plate. It is concluded that these parameters play significant roles on the dynamic behavior of porous MEE-FG plates. Presented numerical results can serve as benchmarks for future analyses of MEE-FG plates with porosity phases.

Nonlinear bending analysis of porous FG thick annular/circular nanoplate based on modified couple stress and two-variable shear deformation theory using GDQM

This is the first attempt to consider the nonlinear bending analysis of porous functionally graded (FG) thick annular and circular nanoplates resting on Kerr foundation. The size effects are captured based on modified couple stress theory (MCST). The material properties of the porous FG nanostructure are assumed to vary smoothly through the thickness according to a power law distribution of the volume fraction of the constituent materials. The elastic medium is modeled by Kerr elastic foundation which consists of two spring layers and one shear layer. The governing equations are extracted based on Hamilton's principle and two variables refined plate theory. Utilizing generalized differential quadrature method (GDQM), the nonlinear static behavior of the nanostructure is obtained under different boundary conditions. The effects of various parameters such as material length scale parameter, boundary conditions, and geometrical parameters of the nanoplate, elastic medium constants, porosity and FG index are shown on the nonlinear deflection of the annular and circular nanoplates. The results indicate that with increasing the material length scale parameter, the nonlinear deflection is decreased. In addition, the dimensionless nonlinear deflection of the porous annular nanoplate is diminished with the increase of porosity parameter. It is hoped that the present work may provide a benchmark in the study of nonlinear static behavior of porous nanoplates.

Galerkin’s approach for buckling analysis of functionally graded anisotropic nanoplates/different boundary conditions

For the first time, buckling behavior of functionally graded (FG) nanoplates made of anisotropic material (beryllium crystal as a hexagonal material) is investigated. Also, it is the first time that the size-dependent behavior of nanostructured systems is studied for buckling response of the graded anisotropic material. The properties of graded material are assumed vary exponentially through the z-direction. Nonlocal strain gradient theory is utilized to predicate the size-dependent buckling behavior of the nanoplate. The nanoplate is modeled by a higher order shear deformation refined plate theory in which any shear correction factor not used. Governing equations and boundary conditions are obtained using a virtual work of variational approach. To solve the buckling problem for different boundary conditions, Galerkin’s approach is utilized. Finally, the influences of different boundary conditions, small-scale parameters, geometry parameters and exponential factor are studied and discussed in detail. It is hoped that the present numerical results can help the engineers and designers to understand and predict the buckling response of FG anisotropic materials.

Thermo-resonance analysis of an excited graphene sheet using a new approach

Forced vibration of graphene nanoplate based on a refined plate theory in conjunction with higher-order nonlocal strain gradient theory in the thermal environment has been investigated. Regarding the higher-order nonlocal strain gradient theory, both stress nonlocality and size-dependent effects are taken into account, so the equilibrium equations which are governing on the graphene sheet have been formulated by the theory. The extracted equations will be solved by a closed-form approach by which maximum dynamic deflection has been obtained while all the edges are considered as simple boundaries. To approve the results of the plate theory, several available well-known references have been compared. Finally, after determining the accuracy of the equations, different parameters have been examined for showing the behavior of the nanoplate exposed to a transverse dynamic harmonic load.

Non-Local Buckling Analysis of Functionally Graded Nanoporous Metal Foam Nanoplates

In this study, the buckling of functionally graded (FG) nanoporous metal foam nanoplates is investigated by combining the refined plate theory with the non-local elasticity theory. The refined plate theory takes into account transverse shear strains which vary quadratically through the thickness without considering the shear correction factor. Based on Eringen’s non-local differential constitutive relations, the equations of motion are derived from Hamilton’s principle. The analytical solutions for the buckling of FG nanoporous metal foam nanoplates are obtained via Navier’s method. Moreover, the effects of porosity distributions, porosity coefficient, small scale parameter, axial compression ratio, mode number, aspect ratio and length-to-thickness ratio on the buckling loads are discussed. In order to verify the validity of present analysis, the analytical results have been compared with other previous studies.

Investigation of influence of homogenization models on stability and dynamic of FGM plates on elastic foundations

In this paper, the effect of the homogenization models on buckling and free vibration is presented for functionally graded plates (FGM) resting on elastic foundations. The majority of investigations developed in the last decade, explored the Voigt homogenization model to predict the effective proprieties of functionally graded materials at the macroscopic-scale for FGM mechanical behavior. For this reason, various models have been used to derive the effective proprieties of FGMs and simulate thereby their effects on the buckling and free vibration of FGM plates based on comparative studies that may differ in terms of several parameters. The refined plate theory, as used in this paper, is based on dividing the transverse displacement into both bending and shear components. This leads to a reduction in the number of unknowns and governing equations. Furthermore the present formulation utilizes a sinusoidal variation of displacement field across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor. Equations of motion are derived from Hamilton’s principle. Analytical solutions for the buckling and free vibration analysis are obtained for simply supported plates. The obtained results are compared with those predicted by other plate theories. This study shows the sensitivity of the obtained results to different homogenization models and that the results generated may vary considerably from one theory to another. Comprehensive visualization of results is provided. The analysis is relevant to aerospace, nuclear, civil and other structures.

Discrepancy on the free vibration of laminated composite plates coupled to a compressible and incompressible fluid domain

This paper examines the free vibration of laminated composite plates in contact with a water-filled cavity. The fluid compressibility is considered in the analysis. The main objective is to evaluate the natural frequencies of the plate-fluid coupled system, compare the results with those obtained assuming an incompressible fluid domain, and examine the influence of the laminate stacking sequence. Classical and refined plate theories are considered in a compact manner. The Ritz method is used to obtain approximations of the kinetic and potential energy of both the fluid and the plate. The accuracy of the formulation is demonstrated by comparing the results to 3D finite element solutions obtained via commercial software. Parametric studies are carried out to examine the influence of the geometry, lamination angles and boundary conditions on the natural frequencies and the error incurred due to neglecting fluid compressibility. The error is higher when thick, stiff plates with rigid supports are considered. The effect of the fluid domain depth on the natural frequencies depends on the symmetric or antisymmetric nature of the mode shape. The orientation of the laminas becomes highly relevant when rectangular plates with mixed supports are considered.

Interfacial Strain Energy Continuity Assumption-Based Analysis of an Orthotropic-Skin Sandwich Plate Using a Refined Layer-by-Layer Theory

A layer-by-layer analysis in combination with a refined plate theory to predict the mechanical behavior of orthotropic-face sandwich structures in a general lamination configuration is suggested. This approach is compatible with uniform deformation principles based on the aspect ratio of the laminate incorporating the continuity of the flexural and in-plane displacements. The interlaminar continuity equations are also constrained using Lagrange multipliers and introducing 14 new variables. The values of the existing and newly introduced variables are determined through the total potential energy minimization technique. The aforesaid numerical analysis is done by the MuPAD code, and the numerical outcomes are compared with other exact reliable published results. The approach proposed is capable of handling the analysis of sandwich structures.

Thermo-mechanical wave dispersion analysis of nonlocal strain gradient single-layered graphene sheet rested on elastic medium

Present research is mainly devoted to show the effects of temperature change on the mechanical properties of in-plane waves propagating in a single-layered graphene sheet. Graphene sheet is assumed to be rested on an elastic medium. Influences of elastic substrate on the behavior of graphene sheet is modeled by the means of a two parameter elastic foundation. In addition, a trigonometric refined plate theory is applied to derive the kinematic relations. Also, a nonlocal strain gradient theory is utilized to show the size-dependency of graphene sheet. In the framework of Hamilton’s principle, the nonlocal governing differential equations are derived. Moreover, an analytical approach is applied to find the unknowns of the final eigen value equation. At the end of the paper, it is tried to clarify the influences of each parameter by the means of some diagrams.

Wave transmission across laminated composite plate in the subsonic flow Investigating Two-variable Refined Plate Theory

Abstract The main aim of this paper is to employ an extension of two-variable Refined Plate Theory (RPT2) to determine Sound Transmission Loss (STL) across laminated composite plate in contrast to external excitation. It should be noted that RPT2 known as a second type of Refined Plate Theory which includes the extension term in displacement field. To achieve this goal, firstly, the lateral displacements are expanded by considering three parts, including shear, bending and extension term to provide an analytical model based on two-variable Refined Plate Theory without considering the effect of the shear correction coefficient in governing equations. Secondly, vibro-acoustic analysis is administered by incorporating the laminated composite plate equation beside the acoustic wave equation, simultaneously. Consequently, the power transmission through the structure is specified due to a plane sound wave. Beside, in order to illustrate the accuracy of the present formulation (RPT2), the obtained STL is compared with those available in literature. However, the application of present study (RPT2) is clarified in the acoustical designs due to presenting more precise results in comparison with Classical Plate Theory. Eventually, the numerical results are achieved to determine the influences of various properties on STL.

Vibration analysis of sandwich plates with carbon nanotube-reinforced composite face-sheets

A layer-wise sandwich model is employed for vibration analysis of symmetric sandwich plates with two thin carbon nanotube-reinforced composite (CNTRC) face-sheets, in which a new polynomial refined plate theory is proposed for the core and the classical plate theory is adopted for the face-sheets. By virtue of symmetry, the in-plane displacements are neglected. By maintaining the displacement continuity conditions at the interfaces, the displacement fields are expressed with only two transverse displacement components, which are given in summations of products of beam characteristic functions in one direction and unknown functions in the other. Then, the multi-term Kantorovich-Galerkin method (MTKGM) is extended to obtain semi-analytical solutions for vibration response of symmetric CNTRC sandwich plates resting on elastic foundation. Non-dimensional natural frequencies are obtained for CNTRC sandwich plates, and the effects of the number of approximate terms, boundary conditions, sandwich configurations, volume fractions of carbon nanotube, plate aspect ratio, core-to-skin thickness ratio, and foundation stiffness are examined. The new polynomial refined plate theory presented provides an improved yet relatively simple model for vibration analysis of sandwich structures.

Thermodynamic effect on the bending response of elastic foundation FG plate by using a novel four variable refined plate theory

ABSTRACTThis article deals with the study of the thermodynamic behavior of functionally graded material plates resting on two-parameter elastic foundation. An analytical solution based on a new shear refined deformation theory is presented. The displacement field used in the present refined theory contains undetermined integral forms and involves only four unknowns to derive. The plate is assumed simply supported and subjected to two different temperatures fields across its thickness. The mechanical characteristics of the plate are assumed to be varied across the thickness according to a simple exponential law distribution. The governing equations and boundary conditions are derived using the principle of virtual displacements and Navier solution technique is adopted to derive analytical solutions. A detailed numerical study of the present new refined theory is carried out to examine the influence of the time’s parameter, foundation’s parameters and deflection on the bending response of the FG plate.

A Modified Kirchhoff plate theory for Free Vibration analysis of functionally graded material plates using meshfree method

In this paper, a modified Kirchhoff theory is presented for free vibration analyses of functionally graded material (FGM) plate based on modified radial point interpolation method (RPIM). The shear deformation effects are taken account into modified theory to ignore the locking phenomenon of thin plates. Due to the proposed refined plate theory, the number of independent unknowns reduces one variable and exists with four degrees of freedom per node. The simulated free vibration results employed by the modified RPIM are compared with the other analytical solutions to verify the effectiveness and the accuracy of the developed mesh-free method. Detail parametric studies of the proposed method are then conducted including the effectiveness of thickness ratio, boundary condition and material inhomogeneity on the sample problems of square plates. Results illustrated that the modified mesh-free RPIM can effectively predict the numerical calculation as compared to the exact solutions. The obtained numerical results are indicated that the proposed method are stable and well accurate prediction to evaluate with other published analyses.

On the equations of the analytical dynamics of the quasi-3D plate theory of I. N. Vekua type and some their solutions

The plate theory of Nth order is constructed on the background of the Lagrangian variational formalism of analytical dynamics of continuum systems and the dimensional reduction approach of I. N. Vekua – A. A. Amosov. The plate model is defined within the configuration space, the set of field variables, and the Lagrangian density. The field variables are determined by the coefficients of the biorthogonal expansion of the spatial displacement vector field with respect to the dimensionless normal coordinate. The dynamic equations are derived as Lagrange equations of the second kind of the two-dimensional continuum system. The dynamics of the plane elastic layer is considered as an example, the normal wave propagation is described on the basis of refined plate theories of various orders, and the convergence of approximate solutions to the exact solution of the three-dimensional elastodynamics problem is analyzed for different wavenumbers.

Geometrically nonlinear analysis of functionally graded material plates using an improved moving Kriging meshfree method based on a refined plate theory

This paper presents a novel formulation based on an improved moving Kriging (iMK) meshfree method and a refined plate theory (RPT) for geometrically nonlinear static and dynamic analyses of functionally graded material (FGM) plates. The nonlinearity of plates is formed in the total Lagrangian approach based on the von Karman strain assumptions. The present plate formulation has the following advantages: only four variables for each node, the tangential stress-zero boundary condition at the top and bottom surfaces of plates, and the stringent continuity requirement of the generalized displacements being simply treated without any additional variables. The novelty of the iMK meshfree method is to propose a quartic spline correlation function for establishment of basic shape functions. Thanks to this improvement, the solution of the iMK meshfree method is stable and no longer depends on an uncertain adjustment of correlation parameter θ. In addition, a simple rotation-free technique originated from isogeometric analysis (IGA) is successfully applied to the iMK meshfree method for enforcing the slopes of clamped boundary condition. Several numerical examples are provided to demonstrate high performance of the present method in comparison with other numerical methods.